The laws of algebra of logic

Modern computers based on "ancient" electronic computers, as the basic principles of work are based on certain postulates. They are called the laws of algebra of logic. For the first time such a discipline was described (of course, not as detailed as in the modern form) by the ancient Greek scholar Aristotle.

Representing a separate section of mathematics, in which the calculus of propositions is studied, the algebra of logic has a number of clearly constructed conclusions and conclusions.

In order to better understand the topic, we will analyze the concepts that will help to learn the laws of algebra of logic in the future.

Perhaps the main term in the discipline being studied is a statement. This is a statement that can not be both false and true. He is always inherent in only one of these characteristics. It is conventionally accepted to assign truth to 1, falsity to 0, and the sentence itself to be called a Latin letter: A, B, C. In other words, the formula A = 1 means that A is true. With statements you can act in a variety of ways. In brief, we will consider the actions that can be taken with them. We also note that the laws of algebra of logic can not be learned without knowing these rules.

1. The disjunction of two statements is the result of the operation "or". It can be either false or true. The symbol "v" is used.

2. Conjunction. The result of such an action, performed with two statements, will be a new statement, true only if both initial statements are true. The operation "and", the symbol "^" is used.

3. The implication. The operation "if A, then B". The result is a statement that is false only if A is true and F is false. The "->" character is used.

4. Equivalence. The operation "A if and only then B, when". This statement is true in cases when both variables have the same estimates. The symbol "<->" is used.

There are also a number of operations close to the implication, but they will not be considered in this article.

Now let us consider in detail the basic laws of the algebra of logic:

1. Commutative or relocatable states that the change of places of logical terms in operations of conjunction or disjunction on the result does not affect.

2. Associative or associative. According to this law, variables in conjunctions or disjunctions can be grouped together.

3. Distributive or distributive. The essence of the law is that the same variables in the equations can be taken out of the brackets, without changing the logic.

4. De Morgan's law (inversion or negation). Denial of the conjunction operation is equivalent to disjunctioning the negation of the original variables. Negation from disjunction, in turn, is equal to the conjunction of negation of the same variables.

5. Double negation. The negation of a certain utterance twice gives as a result the initial statement, three times its denial.

6. The law of idempotency looks like this for logical addition: xvxvxvx = x; For multiplication: x ^ x ^ x ^ = x.

7. The law of non-contradiction says: two statements, if they are contradictory, can not be true at the same time.

8. The law of exclusion of the third. Among the two contradictory statements, one is always true, another is false, the third is not given.

9. The law of absorption can be written in this way for logical addition: xv (x ^ y) = x, for multiplication: x ^ (xvy) = x.

10. Law of gluing. Two adjacent conjunctions are able to glue together, forming a conjunction of a smaller rank. In this case, the variable, according to which the original conjunctions were glued, disappears. Example for logical addition:

(X ^ y) v (-x ^ y) = y.

We have considered only the most commonly used laws of algebra of logic, which in fact can be many more, since often logical equations acquire a long and florid appearance, which can be reduced by applying a number of similar laws.

As a rule, for the convenience of counting and identifying the results, special tables are used. All the existing laws of the algebra of logic, the table for which has a common structure of the grid rectangle, is painted out, distributing each variable into a separate cell. The larger the equation, the easier it is to cope with it using tables.